Distance magic labeling in complete 4-partite graphs

TL;DR

This paper investigates the conditions under which complete 4-partite graphs can be assigned distance magic labelings, proving a conjecture that characterizes such graphs based on a specific inequality involving part sizes.

Contribution

The paper proves a conjecture characterizing when complete 4-partite graphs admit distance magic labelings, extending previous results for smaller values of k.

Findings

Proved the conjecture for k=4.

Extended earlier results from k=2,3 to k=4.

Established a necessary and sufficient condition involving part sizes.

Abstract

Let $G$ be a complete $k$-partite simple undirected graph with parts of sizes $p_1\le p_2...\le p_k$. Let $P_j=\sum_{i=1}^jp_i$ for $j=1,...,k$. It is conjectured that $G$ has distance magic labeling if and only if $\sum_{i=1}^{P_j} (n-i+1)\ge j{{n+1}\choose{2}}/k$ for all $j=1,...,k$. The conjecture is proved for $k=4$, extending earlier results for $k=2,3$.